Higher November 2021 Paper 2 Q22
22 The diagram shows a sphere of diameter \(x\) cm and a pyramid \(ABCDE\) with a horizontal rectangular base \(BCDE\).
Diagram NOT accurately drawn
The vertex \(A\) of the pyramid is vertically above the centre \(O\) of the base so that \(AB = AC = AD = AE\).
\(BC = x\) cm, \(CD = 2x\) cm and \(AO = 5x\) cm.
The volume of the sphere is \(288\pi\) cm3
Calculate the total surface area of the pyramid.
Give your answer correct to the nearest cm2
(6)
| Scheme | Marks |
|---|---|
eg \(\dfrac{4}{3}\pi r^3 = 288\pi\) oe \(\dfrac{4}{3}\pi\left(\dfrac{x}{2}\right)^3 = 288\pi\) oe | M1 |
| \(x = 12\) | A1 |
\(\sqrt{(5 \times \text{‘}{12}\text{’})^2 + (0.5 \times \text{‘}{12}\text{’})^2}\ (= 6\sqrt{101} = 60.299\ldots)\) oe or \((OC =)\ 0.5\sqrt{\text{‘}{24}\text{’}^2 + \text{‘}{12}\text{’}^2}\ (= 6\sqrt{5})\) and \(AC = \sqrt{\text{‘}{(6\sqrt{5})}\text{’}^2 + \text{‘}{60}\text{’}^2}\ (= 6\sqrt{105})\) and \(\sqrt{\text{‘}{(6\sqrt{105})}\text{’}^2 - \text{‘}{12}\text{’}^2}\ (= 6\sqrt{101})\) oe | M1 |
\(\sqrt{(5 \times \text{‘}{12}\text{’})^2 + (1 \times \text{‘}{12}\text{’})^2}\ (= 12\sqrt{26} = 61.188\ldots)\) oe or \((OC =)\ 0.5\sqrt{\text{‘}{24}\text{’}^2 + \text{‘}{12}\text{’}^2}\ (= 6\sqrt{5})\) and \(AC = \sqrt{\text{‘}{(6\sqrt{5})}\text{’}^2 + \text{‘}{60}\text{’}^2}\ (= 6\sqrt{105})\) and \(\sqrt{\text{‘}{(6\sqrt{105})}\text{’}^2 - \text{‘}{6}\text{’}^2}\ (= 12\sqrt{26})\) oe | M1 |
(‘12’ × 2(‘12’)) + 2(0.5 × ‘12’ × ‘\(12\sqrt{26}\)’) + 2(0.5 × 2‘12’ × ‘\(6\sqrt{101}\)’) oe eg ‘288’ + 2 × ‘\(72\sqrt{26}\)’ + 2 × ‘\(72\sqrt{101}\)’ or ‘288’ + 2 × ‘367.129’… + 2 × ‘723.59…’ oe | M1 |
| Correct answer scores full marks (unless from obvious incorrect working) Answer: 2469 | A1 |
| (6) | |
| (6 marks) |
Notes
M1: for using the formula for the volume of a sphere correctly and equating it to \(288\pi\)
M1: (dep on first M1 and using their value for \(x\)) for using Pythagoras to find the perp height of faces CAD or BAE
or
a correct method to find angle CAD or BAE
M1: (dep on first M1 and using their value for \(x\)) for using Pythagoras to find the perp height of faces ABC or AED
or
a correct method to find angle BAC or DAE
M1: (dep on first M1 using their value for \(x\) and correct working for heights of each triangle) for working out the total surface area of the pyramid
A1: 2469 - 2470